Given the equation Square root of 2x plus 1 = 3, solve for x and identify if it is an extraneous solution.

Solution:

Given, the equation is square root of 2x plus 1 = 3

We have to find x and identify if x is an extraneous solution.

An extraneous solution is a root of a transformed equation that is not a root of the original equation because it was excluded from the domain of the original equation.

Example:

Consider the equation (x + 3)/x = 5/x

On cross multiplication,

x(x + 3) = 5x

x² + 3x = 5x

x² + 3x - 5x = 0

x² - 2x = 0

x(x - 2) = 0

So, x = 0

x - 2 = 0

x = 2

The solutions are 0 and 2.

Here, x = 0 is not solution

But x = 2 is the real solution of the equation

So, x = 0 is an extraneous solution.

Now, proceeding with the given problem,

The equation can be written as \(\sqrt{(2x+1)}=3\)

Squaring on both sides,

\((\sqrt{(2x+1)})^{2}=(3)^{2}\)

2x + 1 = 9

2x = 9 - 1

2x = 8

x = 8/2

x = 4

Therefore, the solution is x = 4 and it is not an extraneous root as it satisfies the original solution.

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Given the equation Square root of 2x plus 1 = 3, solve for x and identify if it is an extraneous solution.

Summary:

The solution of the equation Square root of 2x plus 1 = 3 is x = 4 and it is not an extraneous solution.